Mr. Trump’s transition team said Friday that the president-elect received a “very nice letter” from Russian president Vladimir Putin, urging Mr. Trump to act “in a constructive and pragmatic manner” to “restore the framework of bilateral cooperation.”

The letter, dated Dec. 15, also notes that serious global and regional challenges “show that the relations between Russia and the U.S. remain an important factor in ensuring stability and security in the modern world.”

In response, Mr. Trump agreed with Putin, adding that he hopes “we do not have to travel an alternate path.”

Putin spokesman Dmitry Peskov confirmed that Putin sent the letter, “voicing hope for an improvement of bilateral ties,” according to the Interfax news agency. Mr. Trump’s transition team described the text as an unofficial translation.

The exchange comes on the heels of comments by Mr. Trump and Putin about the need to strengthen their countries’ nuclear arsenals. Mr. Trump declared Thursday on Twitter that the U.S. should “greatly strengthen and expand its nuclear capability” until the rest of the world “comes to its senses” regarding nuclear weapons. Those comments echoed an earlier statement by Putin, who said earlier Thursday that strengthening his country’s nuclear capabilities should be a chief military objective in the coming year.

Putin downplayed the significance of Mr. Trump’s comments at a marathon end-of-year news conference Friday. Putin said he sees “nothing unusual” in Mr. Trump’s pledge to strengthen the U.S. nuclear forces, saying the statement is in line with the president-elect’s campaign promises.

In his wide-ranging remarks, the Russian leader claimed that his country’s military is stronger than that of any potential aggressor, but acknowledged that the U.S. military is bigger. He also cast the modernization of Russia’s nuclear arsenal as a necessary response to a U.S. missile defense system.

Mr. Trump extolled Putin’s leadership during the campaign and called for a tempered approach to U.S.-Russia relations. And while Putin has described Mr. Trump in favorable terms, observers say Russia’s interest is centered around winning relief from crippling sanctions implemented under the Obama administration.

The exchange between Mr. Trump and Putin comes as the U.S. president-elect reportedly seemed to welcome an “arms race” in a brief conversation with MSNBC’s Mika Brzezinski Friday morning.

“Let it be an arms race,” he said Friday, according to Brzezinski, co-host of the network’s show Morning Joe.

Brzezinski reported that the president-elect had told her in a telephone call that took place off camera during a commercial break, “We will outmatch them at every pass... and outlast them all.”

Tensions have been mounting between the U.S. and Russia in recent months over accusations by the Obama administration that the Russian government hacked the emails of U.S. citizens and institutions, including political organizations, and handed them over to DCLeaks.com and WikiLeaks for distribution. The FBI said this month that it supports the CIA’s conclusion that Russia interfered in the presidential election with the goal of supporting Mr. Trump.

Mr. Trump has repeatedly dismissed claims that Russia sought to usher in his victory, but his remarks indicate that Russia may not be an exception to Mr. Trump’s vow to assert American influence through “unquestioned military strength.” His comments are prompting fears of a nuclear race in the making if Mr. Trump breaks longstanding nuclear treaties, and if Russia or other countries seek to counter American nuclear gains.

The two countries signed the New START treaty in 2010, capping the number of nuclear warheads and missile launchers each country can possess. The agreement is in effect until 2021.

“Modernization is quite different from expansion,” said Michael McFaul, Obama’s ambassador in Moscow between 2012 and 2014. “Modernization is allowed under the New START treaty; expansion is not.”

“Putin is smartly playing down the tweet because he has several concessions that he wants to secure from President Trump - lifting of sanctions, support for his war in Syria, recognition of Crimea as part of Russia - and therefore does not want to rock the boat right now by reacting to this vague Trump statement,” McFaul added.